Genus 2 curves in isogeny class 4096.b
Label | Equation |
---|---|
4096.b.65536.1 | \(y^2 = x^5 - x\) |
L-function data
Analytic rank: | \(0\) | ||||||||||||||||||||||
Mordell-Weil rank: | \(0\) | ||||||||||||||||||||||
Bad L-factors: |
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Good L-factors: |
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See L-function page for more information |
Sato-Tate group
\(\mathrm{ST} =\) $J(C_2)$, \(\quad \mathrm{ST}^0 = \mathrm{U}(1)\)
Decomposition of the Jacobian
Splits over the number field \(\Q (b) \simeq \) \(\Q(\sqrt{2}) \) with defining polynomial:
\(x^{2} - 2\)
Decomposes up to isogeny as the square of the elliptic curve isogeny class:
Elliptic curve isogeny class 2.2.8.1-64.1-a
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism algebra over \(\Q\):
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-1}) \) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\C\) |
Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\zeta_{8})\) with defining polynomial \(x^{4} + 1\)
Endomorphism algebra over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\mathrm{M}_2(\)\(\Q(\sqrt{-2}) \)\()\) |
\(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\mathrm{M}_2 (\C)\) |
More complete information on endomorphism algebras and rings can be found on the pages of the individual curves in the isogeny class.